MHB Show the following sequence as a monotone increasing

[cbarker1]

Dear Everyone, Here is the sequence: Let $S\subset\Bbb{R}$ and ${x}_{n}\in S$ and $S\ne\emptyset$ . ${x}_{n-1}<{x}_{n}\le\sup S$ for all $n\ge2$. Prove the sequence is monotone increasing.

I need help proving it; I do not know where to start? Thanks
Carter

 

[S.G. Janssens]

I'm confused: Isn't this already given since $x_{n-1} < x_n$ for all $n \ge 2$, assuming $x_1$ is the first term in the sequence?

 

[cbarker1]

The assumption is correct where ${x}_{1}$ is given. So I can say the sequence is monotone increasing?

 

[S.G. Janssens]

Yes. I mean, you are given terms $x_n$ in a set $S$ such that $x_{n-1} < x_n$ for all $n \ge 2$.
That latter inequality is the definition of strict monotonicity, so if the exercise really reads like this, then I cannot see what you would have to do else. (The supremum does not play any role, either.)

Anyone else here on board that sees something I overlooked?